Firm's value of equity vs volatility of the firm

[sl]

Hello,

In my reading I came across what looks like a contradiction between what the two authors, in their respective books, say. I don't know whether I have interpreted them correctly or not.

According to Stulz (Merton model),
Value of equity increases with volatility of the firm

According to Hull (Vol smiles)
When a firm's equity value decreases, the amount of leverage increases, which essentially increases
the volatility of the underlying asset.

Your thoughts please.

 

[David Harper CFA FRM]

Hi laxmsun,

This has been raised before and it seems to me the burden is on Hull's statement, as Hull's statement (The Reason for the Smile in Equity Options) has, in my opinion, not completely survived inspection. But, first, there is a superficial difference:

• Stulz's statement is a pretty rock-solid statement about the Black-Scholes-Merton. It simply says that higher ASSET volatility implies a higher call option price, where in the Merton model, asset volatility (not equity volatility) and asset (firm) price are inputs and the output (c) is the price of equity, as a call option on the firm's assets.
• Hull's statement appears to be about debt-to-equity ("as a company's equity declines in value, the company's leverage increases."). So note a superficial difference: for any given asset and asset volatility, we can have different leverages (debt-to-equity).

But more substantively, Hull's statements is tricker than it seems at first glance:

• It is not clear to me why he can assert "that we can expect the volatility of a stock to be a decreasing function of the stock price and is consistent with Figures 19.3 and 19.4." That is, 19.3 and 19.4 posit an equities smile (smirk) which declines as the STRIKE price increases; ie, volatility decreasing as call option becomes out-of-the-money (OTM) which is higher strike/lower stock ... figure 19.3 appears to show volatility as increasing function of stock price (!?) ... that would resolve except i perceive a deeper problem:

• More to the point (and I've written about this before) I think there is logical fallacy in Hull (i absolutely could be wrong, but nobody has clarified yet why...): he basically says that a company with high leverage should have greater equity volatility. This much is intuitive. But that would impact the whole implied volatility curve (from ITM options on the stock to OTM options on the stock). I don't get how the increased equity volatility implies that ITM call options (or OTM puts) on that equity would have higher volatility that OTM call options on that same equity which is now more volatile (or that, by implication, the left lognormal tail is heavier).

It's a long way to tentatively agree with you, unless somebody else can better interpret Hull.